# The Lefever-Lejeune nonlinear lattice: convergence dynamics and the structure of equilibrium states

## TL;DR

This paper analyzes the Lefever-Lejeune nonlinear lattice model for vegetation growth, identifying stability regimes, convergence behaviors, and the structure of equilibrium states through analytical and numerical methods.

## Contribution

It provides the first detailed stability analysis of the discrete Lefever-Lejeune lattice and links its dynamics to the continuous PDE counterpart.

## Key findings

- Threshold for destabilization depends on lattice parameters
- Existence of non-uniform equilibrium states
- Numerical simulations confirm analytical stability criteria

## Abstract

We consider the Lefever-Lejeune nonlinear lattice, a spatially discrete propagation-inhibition model describing the growth of vegetation densities in dry-lands. We analytically identify parametric regimes distinguishing between decay (associated with spatial extinction of vegetation patches) and potentially non-trivial time-asymptotics. To gain insight on the convergence dynamics, a stability analysis of spatially uniform states is performed, revealing the existence of a threshold for the discretization parameter which depends on the lattice parameters, below which their destabilization occurs and spatially non-uniform equilibrium states may emerge. Direct numerical simulations justified that the analytical stability criteria and parametric thresholds effectively describe the above transition dynamics and revealed the rich structure of the equilibrium set. Connections with the continuous sibling Lefever-Lejeune partial differential equation are also discussed.